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A reader of the Matrices and Determinants chapter in Interactive Mathematics recently wrote and asked where matrices and determinants come from and why do they work?

Matrices are essential for solving large sets of simultaneous equations using a computer. We certainly don’t want to use a different letter for each variable in our problem (or lots of subscripts, like a₃₄) because it would slow down the solution process and would be horrible to code. With matrices, we don’t have to include any variables – just the numbers in front of those variables.

So for example, suppose we are trying to solve this 4×4 system of equations:

3x + 4y + 2z −6w = 5
x − 5y + 7z + 10w = −8
8x + 5y − z + 7w = 8
6x − 4y + 12z + 15w = 4

We only need to give the computer the coefficients, like this:

3   4   2  −6  |  5
1  −5   7  10  | −8
8   5  −1   7  |  8
6  −4  12  15  |  4

The computer just works on the numbers − it doesn’t need the letters.

The Han Chinese and Simultaneous Equations

Here’s a problem from a Chinese mathematics book written in 200BC. (Source)

There are three types of corn, of which three bundles of the first, two of the second, and one of the third make 39 measures. Two of the first, three of the second and one of the third make 34 measures. And one of the first, two of the second and three of the third make 26 measures. How many measures of corn are contained of one bundle of each type?

It looks a lot like the kind of problems in textbooks today, doesn’t it?

The remarkable thing about this problem is the way that the Chinese writer solved it. First, they set up the numbers involved as follows:

 1   2   3
 2   3   2
 3   1   1
26  34  39

(They are using rows where we would use columns. It doesn’t matter.)

The instruction is to…

…multiply the middle column by 3 and subtract the right column as many times as possible, the same is then done subtracting the right column as many times as possible from 3 times the first column. This gives

 0   0   3
 4   5   2
 8   1   1
39  24  39

A similar process occurs to eliminate the 4 in the second row.

 0   0   3
 0   5   2
36   1   1
99  24  39

From this, we can read off the answer for the amount of the 3rd type (99/36 = 11/4) and then substitute to find the second type (17/4) and first type of corn (37/4).

We now call this process Gaussian Elimination after the German mathematician Gauss (1777-1855).

Maybe it should be called Han Elimination.

You can read more interesting history about matrices and determinants from The MacTutor History of Mathematics archive.

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